Interminiband Optical Transitions in Graphene Lateral SuperlatticesYuyu Li and Roberto Paiella*

Department of Electrical and Computer Engineering and Photonics Center, Boston University, 8 Saint Mary’s Street, Boston,Massachusetts 02215, United States

*S Supporting Information

ABSTRACT: Gated graphene superlattices, where in-plane variations in the potential-energy profile are introduced with aperiodic patterning of the gate electrode or dielectric, provide new opportunities for tailoring the electronic and opticalproperties of two-dimensional materials. Here we present a numerical study of the optical transitions between minibandsderived from the same energy band (conduction or valence) in these systems. Giant absorption peaks at voltage-tunable THzfrequencies are obtained, associated with van Hove singularities in the joint density of states of select pairs of minibands.Furthermore, we describe the possibility of interminiband THz gain in the same systems under external carrier injection,resulting from a local population inversion at specific symmetry points of the mini-Brillouin zone, even in the absence of a globalinversion. These results highlight the great potential of engineered graphene superlattices for THz optoelectronic deviceapplications, including modulators, tunable photodetectors, and lasers.

KEYWORDS: graphene, terahertz photonics, superlattices, band structure engineering, optical absorption, optical gain

Solid-state superlattices (SLs), where a periodic potential-energy profile is artificially superimposed on the lattice

potential of a crystalline sample, have been studied extensivelysince the pioneering work of Esaki and Tsu in 1970.1 Thesesystems are most commonly implemented with verticallycoupled semiconductor quantum wells, where the SL potential-energy profile is produced by the spatial variations of theconduction- and valence-band edges along the epitaxial growthdirection. Over the past few decades, these structures haveallowed for the investigation of a wide range of fundamentalelectronic phenomena, such as the opening of new energybandgaps at the minizone boundaries, negative differentialconductance, and Bloch oscillations.2 Their optical propertieshave also been studied in detail, beginning with infraredabsorption spectroscopy measurements.3,4 Similar structureshave then played a key role in the development of mid-infraredand THz quantum cascade lasers, including devices wherestimulated emission involves electronic transitions between thefirst-excited and ground-state SL minibands at the minizoneboundaries.5−7

More recently, renewed interest in SL material and devicephysics has focused on two-dimensional (2D) crystals,particularly graphene, in the presence of an external potential

varying periodically on the sample plane. This lateral SLpotential is obtained naturally in graphene transferred overhexagonal boron nitride (h-BN), when the orientations of thegraphene and h-BN lattices are closely aligned.8−13 Due to thelattice mismatch between the two crystals, a moire pattern withsuitably small periodicity (a few 10 nm) is then produced intheir combined atomic arrangement, and as a result, the energyof the graphene carriers is modulated periodically by thenearby ionized B and N atoms. An alternative approachconsists of introducing periodic variations in the graphenecharge density (leading to a commensurate electrostaticpotential-energy profile) via a suitable patterning of thechemical doping distribution,14 gate electrode,15,16 or gatedielectric.17 The use of strain variations in graphene on aperiodic array of supporting nanospheres has also beeninvestigated recently.18 Compared to moire SLs, theseengineered approaches generally feature more complex devicegeometries but, at the same time, offer greater design flexibilityand, in the case of periodically gated samples, the possibility ofdynamic tunability. Regardless of the physical origin of the

Received: May 4, 2018Published: July 11, 2018

Article

pubs.acs.org/journal/apchd5Cite This: ACS Photonics 2018, 5, 3331−3337

© 2018 American Chemical Society 3331 DOI: 10.1021/acsphotonics.8b00584ACS Photonics 2018, 5, 3331−3337

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periodic potential, the graphene band structure is stronglymodified through the formation of minibands featuring newDirac points on a mini-Brillouin zone (MBZ) determined bythe SL symmetry and periodicity.19−22 The resulting electronicproperties have been studied in detail through extensivetransport measurements, including the observation of fractalband structures under large magnetic fields9−11,17 and ballisticminiband conduction.13 The optical excitation of electron−hole pairs in moire SLs has also been investigated viaabsorption spectroscopy23 and photocurrent measurements.24

In this work, we present a numerical study of the opticaltransitions between minibands derived from the same energyband (conduction or valence) in graphene SLs. While thesetransitions form the basis of the most significant technologicalapplication of semiconductor SLs to date (quantum cascadelasers), they have so far remained largely unexplored in thecontext of 2D materials. Recent theoretical studies have shownthe presence of sharp peaks in the THz conductivity of moire-patterned graphene, associated with resonances across thelowest two valence minibands.25,26 Here, we consider insteadSLs based on periodic in-plane variations of the gate potentialand investigate the dynamic tunability of their THzinterminiband absorption spectra. Furthermore, we describethe feasibility of THz amplification in the same structures andcompute their gain spectra based on a simple model ofinterminiband carrier dynamics. Our simulation results under-score the potential of engineered graphene SLs for thedevelopment of highly integrated and widely tunable THzdevices, including modulators, photodetectors, and lightemitters. It should be noted that graphene in general isalready being widely investigated as a promising THzoptoelectronic material, by virtue of its gapless and linearenergy dispersion, record high room-temperature mobilities,and strongly confined far-infrared plasmonic excitations.Several innovative device concepts leveraging these distinctiveproperties have already been proposed and/or demonstratedfor different applications.27−43 While graphene SLs have notyet been considered in this context, the present work indicatesthat they may also play a particularly significant role for futureTHz device development.

■ PERIODICALLY GATED GRAPHENESUPERLATTICES

The SL geometry under study consists of a single graphenelayer (possibly encapsulated in h-BN for enhanced mobility44)featuring a triangular periodic array of circular regions wherethe electrostatic potential is held at a constant value USLrelative to the remaining area of the sample (Figure 1a). Thisarrangement can be obtained in a double-gated graphene field-effect transistor, by introducing the same periodic variations ineither top or bottom gate potential (with the other gate used toindependently control the graphene Fermi level). Specificdevice structures could be implemented with a periodicpatterning of the top gate contact15,16 or by introducing acommensurate lattice of air holes in the bottom gatedielectric.17 In passing, we note that similar structures (albeitwith larger periods) have also been developed recently for thestudy of THz plasmon polaritons in graphene.45,46 In thefollowing we present simulation results for a constant arrayperiod Λ = 50 nm, which is well below the collision mean freepath of carriers in high-quality graphene samples, and at thesame time large enough to be accessible with currentnanofabrication techniques. The diameter D of the circularregions is fixed at Λ/2.The SL miniband structure is computed by diagonalizing the

electronic effective Hamiltonian near the Dirac points,including the SL periodic potential, in the basis of the energyeigenstates of graphene in the absence of any externalperturbation.19 The absorption spectrum can then be evaluatedfrom the dynamic conductivity using the approach described,for example, in ref 47, which, when applied to a graphene SL,produces the following expression

∑ ∑α ωω

ψ σ ψ

ω δ

=ϵ

|⟨ | | ⟩|

×−

ℏ + − [ − ]

≠

q vc A

f E f Ei E E

k kk k

( ) Im4

( ( )) ( ( ))( ) ( ) ( )

m n mm x n

n m

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2

Ä

Ç

ÅÅÅÅÅÅÅÅÅÅÅÅ É

Ö

ÑÑÑÑÑÑÑÑÑÑÑÑ (1)

Here α is the dimensionless absorption coefficient normalizedto the inverse thickness of a graphene single layer; the incidentlight is assumed to be a harmonic plane wave of angularfrequency ω and linear polarization along the x direction;

Figure 1. Periodically gated graphene SLs. (a) Schematic SL geometry. The circular regions are held at a constant electrostatic potential USLrelative to the surrounding areas. (b) MBZ of the SL of (a). (c) Miniband structure of the same SL for Λ = 50 nm, D = Λ/2, and USL = 73 meV.(d) Dispersion of the minibands of (c) along selected symmetry lines of the MBZ. The solid double arrow indicates the interminiband transitionsunder study in this work. The dashed vertical arrow denotes the excitation of holes in miniband V3 at the pump-light photon energy used in thegain calculations of Figure 4. The curved arrows illustrate intraminiband hole relaxation processes in minibands V2 and V3.

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En(k) and ψn,k are the electronic energy eigenvalues andeigenvectors as a function of crystal wavevector k andminiband index n; the k sum is over the entire MBZ (shownschematically in Figure 1b); the indexes m and n in the secondsum run over all SL minibands (both conduction and valence);vF ≈ 1 × 106 m/s is the graphene Fermi velocity, A is thesample area, σx is the x component of the Pauli matrices, f isthe electronic occupation probability, and τsc = δ−1 is thescattering lifetime. The latter parameter is taken to be 1 ps,corresponding to a carrier mean free path l = vFτsc of about 1μm, which can be readily achieved even at room temperaturein high-quality graphene samples fully encapsulated in h-BN,44

even in the presence of an external SL potential.17

Figure 1c,d shows the calculated electronic band structurenear the K Dirac point of plain graphene (relabeled Γ in theseplots), for the device geometry of Figure 1a with USL = 73meV. As expected in the presence of the periodic SL potential,both valence and conduction bands evolve into a series ofminibands, labeled Vn and Cn in the following (where thepositive integer n increases with increasing energy separationfrom the Dirac point). Importantly, any two consecutiveminibands in these plots are connected to each other at least atone symmetry point of the MBZ, so that no full minigap isopened (as would be the case in a standard 2D electron gasunder similar periodic potentials). This behavior is aconsequence of the chiral nature of the graphene carriers,combined with the inversion and time-reversal symmetry oftheir Hamiltonian,19 which is preserved even in the presence ofthe periodic potential of Figure 1a. (In contrast, full minigapscan be opened in graphene/h-BN moire SLs, where theinversion symmetry of the graphene unit cell is broken by thedifferent potentials of the nearby B and N atoms.11) In anycase, pronounced local minigaps between consecutive mini-bands are still observed in Figure 1c,d at or near othersymmetry points of the MBZ. The miniband structure shownin these figures also features a clear electron−hole asymmetry,which is related to the polarity of the SL voltage and thus canbe reversed by changing the sign of USL.

■ INTERMINIBAND ABSORPTIONDepending on the carrier distribution among the differentminibands, large resonant peaks in the SL absorption spectrumcan be obtained, associated with van Hove singularities in theinterminiband joint density of states. Here we focus on thelocal minigap between the second and third valence minibandsat the X points of the MBZ (indicated by the solid doublearrow in Figure 1d). This minigap occurs between themaximum of the miniband below and a local maximum(with a nearby saddle point) of the miniband above, and as aresult can produce a particularly strong peak in the jointdensity of states, and thus in the interminiband absorptionspectrum. This expectation is confirmed by the simulationresults of Figure 2, where the absorption coefficient α of eq 1 iscomputed under conditions of thermal equilibrium at differenttemperatures T. The graphene Fermi energy EF in thesecalculations is fixed at the bottom of the minigap underconsideration, so as to maximize the electronic populationdifference between minibands V3 and V2 near the X point. Asharp absorption peak at the minigap frequency of 3.7 THz isobtained, with maximum value at low temperature as large as4.7. This value is quite remarkable, considering the ultrasmallthickness of single-layer graphene. The corresponding single-pass absorbance (without accounting for reflection at the SL

surface) is 1 − e−α = 99.1%, as opposed to 2.3% for the case ofinterband transitions in plain graphene.47 As the temperature isincreased, the peak absorbance decreases due to the thermalexcitation of electrons across the V2−V3 minigap, but a largevalue above 60% is still computed at room temperature.Relatively strong and sharp absorption peaks are alsocomputed for the same SL geometry with shorter scatteringlifetimes τsc on the order of a few hundred fs (see SupportingInformation, Figure S1). In passing, we note that eq 1 does notinclude the effect of free-carrier absorption, which however canbe expected to be quite small at the low carrier density of thesample under study (3.3 × 1011 cm−2). For example, using theDrude formula for the free-carrier absorption coefficient ofplain graphene,48 we find a negligible contribution of αFCA =0.006 at the peak absorption frequency of Figure 2.The absorption peak just described can also be tuned

dynamically by varying the SL potential USL (Figure 3a), dueto the resulting variations in the SL miniband structure. Inparticular, the energy separation between minibands V2 andV3 increases with increasing USL, while at the same time theircurvatures near the X point decrease (Figure 3b), leading to aproportional enhancement in their joint density of states. As aresult, the absorption peak is blue-shifted, and simultaneouslyincreased and broadened. This behavior is illustrated in thecolor map of Figure 3a, where we plot the THz-rangeinterminiband absorption spectrum of the structure of Figure 2(again, with Fermi level fixed at the top of miniband V3), fordifferent values of USL at T = 5 K. The frequency of maximumabsorption νpeak, indicated by the dotted line in the color map,can be varied across a wide portion of the THz spectrum withrelatively small changes in SL potential. The progressiveflattening of both minibands V2 and V3 with increasing USLalso explains the observed saturation in νpeak at large values ofthe SL potential. The accessible tuning range (νpeak ≤ 9.2 THzin Figure 3a) can be increased using SLs of shorter period Λ(see Supporting Information, Figure S2), where the larger sizeof the MBZ allows for larger energy variations across eachminiband at any given voltage.By virtue of their large peak values, relatively steep edges,

and broad tunability, the absorption spectra of the SL justdescribed are particularly attractive for the development ofTHz modulators capable of providing high contrast ratios. Toillustrate, Figure 3c shows the single-pass transmission e−α ofthe same structure at different frequencies plotted as a functionof USL. At frequencies above 3.7 THz, e

−α can be decreased bya factor of over 100× by varying the SL potential from 0 to avalue on the order of 100 meV or less. At lower frequencies,

Figure 2. Interminiband absorption in graphene SLs. Absorptionspectrum of the SL of Figure 1 under conditions of thermalequilibrium at different temperatures, with Fermi level at the top ofminiband V3.

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the maximum achievable contrast ratios are smaller, due to theproportionally smaller absorption peaks, but can be increasedusing SLs with larger period (see Supporting Information,Figure S2). The underlying modulation dynamics can also beexpected to be ultrafast, since it depends on the externalperturbation of an electronic band structure (e.g., similar to thequantum-confined Stark effect in semiconductor quantumwells49) and, thus, is not limited by any charge transportphenomena. The same interminiband transitions could also beemployed for THz photodetection, even though they do notinvolve the direct generation of new electron−hole pairs.Instead, the energy of the absorbed light in these devices canbe converted into heat and then detected through thephotothermoelectric effect−a process that has already beenshown to be quite efficient in the context of intraband THzabsorption in plain graphene.40 In this context, the narrowbandnature and dynamic tunability of the interminiband transitionsunder study are also quite interesting as a way to enable novelfunctionalities, such as the development of extremelyminiaturized THz spectrometers.

■ INTERMINIBAND GAIN

Next, we investigate THz light emission and amplification inthe same graphene SLs. If the sample is undoped and asignificant number of holes is then introduced in miniband V3(e.g., by optical pumping or electrical injection), light emissionpeaked at the V2−V3 minigap frequency can be expected. Thepossibility of optical gain at the same frequency is enabled bythe separation in reciprocal space between the absolutemaxima of minibands V3 and V2, which are located at the Xand M symmetry points of the MBZ, respectively (Figure 1d).In traditional SLs based on semiconductor quantum wells,when excess electrons (holes) are injected into a miniband,they tend to quickly relax into the available states near theminimum (maximum) of the same miniband, before eventuallydecaying into other minibands at lower (higher) energy.6 Theunderlying intraminiband relaxation mechanism is primarilycarrier−carrier scattering, which is particularly effective atequilibrating carriers among energetically adjacent states (asopposed to states separated by a finite energy gap). Thisgeneral idea is exploited in SL quantum cascade lasers,5−7

where a local population inversion is established between theextrema of two consecutive minibands even without a globalinversion across the same two minibands. A detailedinvestigation of the corresponding intra- and interminibandcarrier relaxation dynamics in graphene SLs has not beenreported yet. However, a similar behavior as just described can

be expected based on time-resolved studies with plaingraphene.50−53 In these reports, intraband equilibration bycarrier−carrier scattering was found to occur on a much fastertime scale compared to interband recombination (about 100 fsvs 1−10 ps at room temperature). As a result, separate quasi-Fermi distributions are established in different bands undernonequilibrium conditions. The associated carrier temperaturedynamics (i.e., cooling to the lattice temperature via opticalphonon emission) is more complex as it depends on the energyrange of the initial hot-carrier distribution,53 but generallytakes place on an intermediate time scale.With such relaxation dynamics in the SL of Figure 1, holes

injected into minibands V3 and V2 will rapidly relax into statesnear their respective maxima at the X and M points, so that alocal population inversion near X can be established. Toestimate the resulting optical gain, we consider an opticalpumping scheme where electrons and holes are introduced inan undoped SL through the absorption of externally incidentlight. The hole densities in minibands V2 and V3 (P2 and P3,respectively) are computed with a simple rate-equation model,similar to what is commonly used to study the thresholdcondition in quantum cascade lasers:54

τ

τ τ

= − + Φ

= − + Φ

dPdt

Pr

dPdt

P Pr

3 3

33 p

2 3

32

2

22 p

l

m

oooooooo

n

oooooooo (2)

In these equations, Φp is the photon flux of the pump light, rn(for n = 2 or 3) is the fraction of incident pump photons thatare absorbed through the creation of holes in miniband Vn,and τn is the lifetime of the corresponding miniband with 1/τ3= 1/τ32 + 1/τ31 and 1/τ2 = 1/τ21, where 1/τnm is the holeinterminiband decay rate from Vn to Vm. The hole densities P2and P3 are obtained by solving eq 2 in steady state (i.e., with d/dt→ 0) and then used to evaluate the quasi-Fermi energies EF2and EF3 of their respective minibands, again under theassumption of ultrafast intraminiband equilibration. Finally,the gain spectrum g(ω) = −α(ω) is computed from eq 1 withthe occupation probabilities f(En(k)) given by quasi-Fermidistribution functions with Fermi energies EFn.In these calculations, the pump wavelength λp is selected so

as to maximize r3, while at the same time keeping r2 as small aspossible (see Supporting Information, Figure S3). Specifically,we use a resonance in the joint density of states of minibandsV3 and C1 (illustrated by the dashed arrow in Figure 1d),which occurs at λp = 11.7 μm under the SL bias conditions of

Figure 3. Dynamic tunability of interminiband absorption in graphene SLs. (a) Absorption spectra of the SL of Figure 2 at 5 K for different valuesof the potential USL. The dotted line indicates the frequency of peak absorption νpeak as a function of USL. For each value of USL, the graphene Fermilevel is fixed at the top of miniband V3. (b) Valence miniband structure of the same sample geometry for different values of USL. (c) Single-passtransmission e−α of the same device at different frequencies plotted as a function of USL.

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Figure 1 (USL = 73 meV). The corresponding values of r3 andr2, evaluated using eq 1, are 10.7% and 2.6%, respectively. Thephoton flux Φp is computed for an input pump intensity(transmitted inside the graphene sheet) equivalent to 2 mWfocused on a 10 × 10 μm2 area. Finally, the scattering lifetimesτnm are taken to be in the picosecond range, in accordance withthe experimental interband relaxation times of plaingraphene.50−53 For simplicity, we also introduce theassumption that these lifetimes are all equal to one another,that is, τ32 = τ31 = τ21 ≡ τ*, and then study the gain-spectrumdependence on the single parameter τ*. In fact, τ21 can beexpected to be shorter than both τ32 and τ31, given the closeenergetic proximity of minibands V2 and V1 along the M−Xdirection in the MBZ (Figure 1d), which favors fast holeinterminiband relaxation from V2 to V1 by carrier−carrierscattering. Therefore, this assumption of equal interminibanddecay rates can possibly lead to overestimating P2 and thusunderestimating the gain coefficient in the simulations below.The interminiband gain spectrum of the SL of Figure 1 is

shown in Figure 4a, computed for T = 5 K and different valuesof the time constant τ* ranging from 1 to 5 ps. In thesecalculations, the broadening parameter δ of eq 1 is set equal to1/τsc + (1/τ2 + 1/τ3)/2,

55 where the second term accounts forthe additional interminiband scattering under the presentnonequilibrium conditions. In each trace of Figure 4a, a sharpfeature centered near the V2−V3 minigap frequency isobtained, with peak value as large as 0.8 for the longest timeconstant. If the same SL is placed inside a vertical opticalcavity, the corresponding amplification factor per round trip e2g

is therefore nearly 5×. The inset shows the hole densities P2and P3 computed in the same simulations. Regardless of thespecific value of τ*, the SL does not support a globalpopulation inversion between the minibands under study (i.e.,P3 is always smaller than P2). The observed gain is thenproduced by the aforementioned local inversion near the Xpoint of the MBZ, where miniband V2 is well below itsmaximum and therefore features smaller hole occupationprobability (i.e., more electrons) than the maximum of V3 atthe same point. Calculations based on the same model justpresented with the same set of values for τ* suggest that thepresence of gain may be possible even at room temperature(see Supporting Information, Figure S4). However, moredefinitive predictions in this respect will require a deeperknowledge of the intra- and interminiband relaxation dynamicsof graphene SLs than presently available.The frequency of peak gain is also tunable with the SL

potential USL, as illustrated in Figure 4b for τ* = 3 ps. Asalready described above, increasing USL has the effect of

flattening both minibands V2 and V3 near the X point and thusincrease their energy separation, and as a result the gainspectrum is shifted to higher frequencies. The flattening of V3throughout the MBZ at large values of USL (e.g., see bottompanel of Figure 3b) also causes a redistribution of its holepopulation across the entire miniband. Under these conditions,absorption of the pump light via transitions involving V3 canbe partially saturated. This effect is not accounted for in themodel of eqs 2, and its inclusion would require furtherassumptions about the SL intra- and interminiband relaxationdynamics. For this reason, in Figure 4b we only consider valuesof USL for which the hole occupation probability of the V3states involved in the pumping transitions remains sufficientlysmall (<0.1%), so that absorption saturation of the pump lightis not a concern. Similar to the case of the absorption spectradescribed above (see Supporting Information, Figure S2), thetuning range of the gain peak can also be extended using SLs ofsmaller period.

■ CONCLUSION

The simulation results presented in this work highlight thegreat potential of periodically gated graphene SLs for THzoptoelectronic device applications. In particular, by virtue oftheir strong van Hove singularities in the joint density of statesof select miniband pairs, these systems can provide narrow-band absorption (or gain under suitable pumping), with giantpeak values for 2D materials. These absorption and emissionfeatures are also broadly tunable (with relatively small changesin the SL potential), which represents a distinctive advantageover traditional THz optoelectronic materials. Specific deviceapplications that can be envisioned include modulators withlarge contrast ratio and high speed, and narrowband tunablephotodetectors. Furthermore, the ability of the same systemsto provide interminiband gain may open the way for a newclass of compact THz lasers inspired by prior work withquantum-well SLs.5−7 The ultimate performance capabilities ofsuch THz sources depend on the detailed intra- andinterminiband carrier dynamics of graphene SLs, and thepresent work therefore provides a direct motivation for theinvestigation of these phenomena. It should also be noted that,while the simulations of Figure 4 have focused on opticalpumping for simplicity, suitable electrical-injection schemescould also be devised, for example, based on tunneling from anadjacent graphene sheet across an intervening 2D insulatorsuch as h-BN. Altogether, graphene SLs therefore represent apromising new materials platform to address a long-standing

Figure 4. Interminiband gain in optically pumped graphene SLs. (a) Low-temperature gain spectrum of the SL of Figure 1 for different values ofthe interminiband relaxation lifetime τ*. Inset: hole densities of minibands V2 and V3 vs τ*. (b) Gain spectra of the same sample geometry for τ* =3 ps and different values of the SL potential USL.

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gap in optoelectronic science and technology, that is, the lackof high-performance solid-state THz devices.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acsphoto-nics.8b00584.

Additional simulation results showing interminibandabsorption versus carrier scattering lifetime and SLperiod, optimization of the pump wavelength used in thegain calculations, and interminiband gain versus temper-ature (PDF).

■ AUTHOR INFORMATIONCorresponding Author*E-mail: rpaiella@bu.edu.ORCIDRoberto Paiella: 0000-0002-7183-6249NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was partly supported by the National ScienceFoundation under Grant No. DMR-1308659. Some of thesimulations were performed using the Shared ComputingCluster facility at Boston University.

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